Summary
The formulation of general shell elements using the method of mixed interpolation of tensorial components (MITC) is reviewed. In particular three elements that were formulated using the MITC method are examined: the MITC4 and MITC8 that were developed for general nonlinear analysis under the restriction of small strains and the MITC4-TLH that was developed for finite strain elasto-plastic analysis of shells.
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References
Ahmad, S., Irons, B.M., and Zienkiewicz, O.C. (1970), “Analysis of thick and thin shell structures by curved finite elements”,Int. J. Num. Meth. Engng.,2, pp. 419–451.
Zienkiewicz, O.C. and Taylor, R.L. (1989), “The Finite Element Method”. (Fourth Edition), Mc Graw Hill.
Reissner, E. (1945), “The effect of transverse shear deformation on the bending of elastic plates”,J. Appl. Mech.,12, pp. 69–76.
Mindlin, R.D. (1951), “Influence of rotatory inertia and shear in flexural motions of isotropic elastic plates”,J. Appl. Mech.,18, pp. 31–38.
Bathe, K.J. and Bolourchi, S. (1979), “A geometric and material nonlinear plate and shell element”,Comput. & Struct.,11, pp. 23–48.
Bathe, K.J. (1982), “Finite Element Procedures in Engineering Analysis” Prentice-Hall, Englewood Cliffs, New Jersey.
Ramm, E. (1977), “A plate/shell element for large deflections and rotations” inFormulations and Computational Algorithms in Finite Element Analysis, (Eds. K.J. Bathe et. al.), M.I.T. Press.
Kråkeland, B. (1978), “Nonlinear analysis of shells using degenerate isoparametric elements”, inFinite Elements in Nonlinear Mechanics, Tapir Publishers.
Dvorkin, E.N. (1984),On Nonlinear Finite Element Analysis of Shell Structures, Ph.D. Thesis, Dept. of Mech. Eng., M.I.T., Cambridge, Mass., U.S.A.
Dvorkin, E.N. and Bathe, K.J. (1984), “A continuum mechanics based four node shell element for general nonlinear analysis”,Engrg. Comput.,1, pp. 77–88.
Bathe, K.J. and Dvorkin, E.N. (1985), “A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation”,Int. J. Num. Meth. Engrg.,21, pp. 367–383.
Bathe, K.J. and Dvorkin, E.N. (1986), “A formulation of general shell elements—the use of mixed interpolation of tensorial components”,Int. J. Num. Meth. Engrg.,22, pp. 697–722.
Dvorkin, E.N. (1992), “On nonlinear analysis of shells using finite elements based on mixed interpolation of tensorial components”, inNonlinear Analysis of Shells by Finite Elements, (Ed. F.G. Rammerstorfer), Springer Verlag.
Bathe, K.J., Dvorkin, E.N. and Ho, L.W. (1983), “Our discrete Kircchoff and isoparametric shell elements for nonlinear analysis-an assessment”,Comput. & Struct.,16, pp. 89–98.
Belytschko, T., Stolarski, H., Liu, W.K., Carpenter, N. and Ong, J. (1985), “Stress projection for membrane and shear locking in shell finite elements”,Comput. Methods Appl. Mech. Engrg.,51, pp. 221–258.
Zienkiewicz, O.C., Taylor, R.L. and Too, J.M. (1971), “Reduced integration techniques in general analysis of plates and shells”,Int. J. Num. Meth. Engrg.,3, pp. 275–290.
Hughes, T.J.R. (1987), “The Finite Element Method” Prentice-Hall, Englewood Cliffs, NJ.
Malkus, D.S. and Hughes, T.J.R. (1978), “Mixed finite element methods in reduced and selective integration techniques: a unification of concepts”,Comput. Methods Appl. Mech. Engrg.,15, pp. 63–81.
Grisfield, M.A. (1986), “Finite Elements and Solution Procedures for Structural Analysis. Vol. I: Linear Analysis”, Pineridge Press, Sivansea.
Belytschko, T., and Tsay, C.S. (1983), “A stabilization procedure for the quadrilateral plate element with one-point quadrature”,Int. J. Num. Meth. Engrg.,19, pp. 405–419.
Belytschko, T. and Leviathan, I. (1994), “Projection schemes for one-point quadrature shell elements”,Comput. Methods Appl. Mech. Engrg.,115, pp. 277–286.
Irons, B.M. and Razzaque, A. (1972), “Experience with the patch test for convergence of finite elements”, inThe Mathematical Foundations of the Finit Element Method with Applications to Partial Differential Equations, (Ed. A.K. Aziz), Academic Press.
Dvorkin, E.N., Pantuso, D. and Repetto, E.A., “A formulation of the MITC4 shell element for finite strain elasto-plastic analysis”,Comput. Methods Appl. Mech. Engrg. (in press).
Strang, G. and Fix, G.J. (1973), “An Analysis of the Finite Element Method”, Prentice Hall.
Marsden, J.E. and Hughes, T.J.R. (1983), “Mathematical Foundations of Elasticity”, Prentice-Hall, Englewood Cliffs, New Jersey.
Dvorkin, E.N., Goldschmit, M.B., Pantuso, D. and Repetto, E.A. (1994) “Comentarios sobre algunas herramientas utilizadas en la resolución de problemas no-lineales de mecánica del continuo”,Rev. Int. de Met. Numéricos para Cálc. y Diseño en Ingeniería,10, pp. 47–65.
Truesdell, C. and Noll, W. (1965), “The nonlinear field theories of mechanics”, inEncyclopedia of Physics, Springer, Berlin.
Malvern, L.E. (1969), “Introduction to the Mechanics of a Continuous Medium” Prentice-Hall, Englewood Cliffs, New Jersey.
Simo, J.C. and Taylor, R.L. (1985), “Consistent tangent operators for rate-independent plasticity”,Comput. Methods Appl. Mech. Engrg.,48, pp. 101–118.
Simo, J.C. and Taylor, R.L. (1986), “A return mapping algorithm for plane stress elastoplasticity”,Int. J. Num. Meth. Engrg.,22, pp. 649–670.
Ortiz, M. and Simo, J.C. (1986), “An analysis of a new class of integration algorithms for elastoplastic constitutive relations”,Int. J. Num. Meth. Engrg.,23, pp. 353–366.
Bathe, K.J. and Cimento, A.P. (1980), “Some practical procedures for the solution of nonlinear finite element equations”,Comput. Methods Appl. Mech. Engrg.,22, pp. 59–85.
Bathe, K.J. and Dvorkin, E.N. (1983), “On the automatic solution of nonlinear finite element equations”,Comput. & Struct.,17, pp. 871–879.
Dvorkin, E.N., Oñate, E. and Oliver, J. (1988), “On a non-linear formulation for curved Timoshenko beam elements considering large displacement/rotation increments”,Int. J. Num. Meth. Engrg.,26, pp. 1597–1613.
Argyris, J. (1982), “An excursion into large rotations”,Comput. Methods Appl. Mech. Engrg.,32, pp. 85–155.
Lee, E.H. and Liu, D.T. (1967), “Finite strain elastic-plastic theory with application to plane-wave analysis”,J. Appl. Phys.,38, pp. 17–27.
Lee, E.H. (1969), “Elastic plastic deformation at finite strain”,J. Appl. Mech.,36, pp. 1–6.
Argyris, J.H. and Doltsinis, J. St. (1979), “On the large strain inelastic analysis in natural formulation. Part I. Quasistatic problems”,Comput. Methods Appl. Mech. Engrg.,20, pp. 213–241.
Argyris, J.H. and Doltsinis, J. St. (1980), “On the large strain inelastic analysis in natural formulation. Part II. Dynamic problems”,Comput. Methods Appl. Mech. Engrg. 21, pp. 91–126.
Simo, J.C. and Ortiz, M. (1985), “A unified approach to finite deformation elastoplasticity based on the use of hyperelastic constitutive equations”,Comput. Methods Appl. Mech. Engrg.,51, pp. 177–208.
Simo, J.C. (1988), “A. Framework for finite strain elasto-plasticity based on maximun plastic dissipation and the multiplicative decomposition, part I: Continuum formulation”,Comput. Methods Appl. Mech. Engrg.,66, pp. 199–219.
Simo, J.C. (1988), “A. Framework for finite strain elasto-plasticity based on maximun plastic dissipation and the multiplicative decomposition, part II: Computational aspects”,Comput. Methods Appl. Mech. Engrg.,68, pp. 1–31.
Hill, R. (1950), “The Mathematical Theory of Plasticity”, Oxford University Press.
Dvorkin, E.N., Pantuso, D. and Repetto, E.A. (1994), “A finite element formulation for finite strain elasto-plastic analysis based on mixed interpolation of tensorial components”,Comput. Methods Appl. Mech. Engrg.,114, pp. 35–54.
Hill, R. (1978), “Aspects of invariance in solid mechanics”,Advances in Appl. Mechs.,18, pp. 1–75.
Athuri, S.N. (1984), “Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations for computational analysis of finitely deformed solids, with applications to plates and shells. Part I”,Comput. & Struct.,18, pp. 93–116.
Krauss, H. (1967), “Thin Elastic Shells”, John Wiley & Sons, New York.
Rodal, J.J.A. and Witmer, E.A. (1979), “Finite-strain large-deflection elastic-viscoplastic finite-element transient analysis of structures”, NASA CR 159874.
Hughes, T.J.R. and Carnoy, E. (1983), “Nonlinear finite element shell formulation accouting for large membrane st rains”,Comput. Methods Appl. Mech. Engrg,39, pp. 69–82.
Simo, J.C. and Fox, D.D. (1989), “On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization”,Comput. Methods Appl. Mech. Engrg.,72, pp. 267–304.
Simo, J.C. Fox, D.D. and Rifai, M.S. (1989), “On a stress resultant geometrically exact shell model. Part II: The linear theory; Computational aspects”,Comput. Methods Appl. Mech. Engrg.,73, pp. 53–92.
Simo, J.C. Fox, D.D. and Rifai, M.S. (1990), “On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory”,Comput. Methods Appl. Mech. Engrg.,79, pp. 21–70.
Simo, J.C. Fox, D.D. and Rifai, M.S. (1990), “On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through-the-thickness stretching”,Comput. Methods Appl. Mech. Engrg.,81, pp. 91–126.
Simo, J.C. and Kennedy, J.G. (1992), “On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity formulation and integration algorithms”,Comput. Methods Appl. Mech. Engrg.,96, pp. 133–171.
Simo, J.C. and Marsden, J.E. (1984), “On the rotated stress tensor and the material version of the Doyle Ericksen formula”,Arch. Rat. Mechs. Anal.,86, pp. 213–231.
Lubliner, J. (1990), “Plasticity Theory”, Macmillan.
Luenberger, D.G. (1984), “Linear and Nonlinear Programming”, Addison-Wesley, Reading, MA.
Eterovic, A.L. and Bathe, K.J. (1990), “A hyperelastic based large strain elasto-plastic constitutive formulation with combined isotropic/kinematic hardening using the logarithmic stress and strain measures”,Int. J. Num. Meth. Engrg.,30, pp. 1099–1114.
Ortiz, M. and Simo, J.C. (1986), “An analysis of a new class of integration algorithms for elastoplastic constitutive relations”,Int. J. Num. Meth. Engrg.,23, pp. 353–366.
Simo, J.C. and Taylor, R.L. (1985), “Consistent tangent operators for rate-independent plasticity”,Comput. Methods Appl. Mech. Engrg.,48, pp. 101–118.
Simo, J.C. and Taylor, R.L. (1986), “A return mapping algorithm for plane stress elastoplasticity”,Int. J. Num. Meth. Engrg.,22, pp. 649–670.
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In memoriam of Juan Carlos Simo.
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Dvorkin, E.N. Nonlinear analysis of shells using the MITC formulation. ARCO 2, 1–50 (1995). https://doi.org/10.1007/BF02904994
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DOI: https://doi.org/10.1007/BF02904994